
Class LB if.ir.r 

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METHODS 



ELEMENTARY SCHOOL STUDIES 



£&££& 



METHODS 



IN 



ELEMENTARY SCHOOL STUDIES 



A BRIEF OUTLINE 



BY 



BERNARD CRONSON, A.B., Pd.D. 

PRINCIPAL PUBLIC SCHOOL NO. 125, BOROUGH OF MANHATTAN 
CITY OF NEW YORK 



THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO. Ltd. 
1906 

All rights reserved 



LIBRARY of CONGRESS 
Two Cooies Received 

JAN 2 1906 

Couyrieht Entry 
let t* . A . / ? 4 £ 
&LASS ^L XXc. No, 
/ 3 ¥ °f d H- 

COPY B. 



Copyright, 1005, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped. Published December, 1905. 



Nortoooti ^rcss 

J. S. Cushing & Co. — Berwick & Smith Co. 

Norwood, Mass., U.S.A. 



CONTENTS. 



I. 


Reading 


II. 


Dictation . 


\ 


Spelling, Punctu 




TALS, ETC. 


III. 


Composition 


IV. 


Grammar . 


V. 


Literature 


VI. 


Nature Study . 


vi r. 


Geography 


VIII. 


History 


IX. 


Civics . , 


X. 


Arithmetic 



ation, Paragraphing, Capi 



•AGE 
11 

24 



41 

63 

76 

87 

95 

103 

112 

120 



"Theory alone is inspiring, but vague; practice 
alone is definite, but formal ; theory combined with 
practice is inspiring, rational, and definite." 



PREFACE. 

A knowledge of specific methods is a necessary 
preparation to the art of teaching, because — 

1. Method is the medium between teacher and pupil. 
It is not the question whether the teacher understands 
the rationale of the processes that concerns the child, 
but the validity or the falsity of the method employed. 
Teaching, or the process whereby one mind causes 
knowledge to arise in another, is an art; the teacher, 
during the performance of his task, is an artisan ; and 
as such he should know how to use his tools, namely, 
his methods. 

2. The mere study of the science of education does 
not of itself insure the mastery of methodology any 
more than theoretic knowledge in any other field car- 
ries with it also skill in application. Even Ruskin, 
with all his keen insight, would scarcely be regarded 
as a competent teacher of painting. And why not? 
" There is in the master's profession," observes J. G. 
Fitch, "the same difference which is observable in all 
other human employments between the skilled and the 
unskilled practitioner, and . . . this difference depends 
in a large measure on a knowledge of the best rules 



8 PREFACE. 

and methods which have to be used. It is easy to say 
of the schoolmaster, 4 nascitur no a fit, 1 and to give this 
as a reason why all training and study of method are 
superfluous. But we do not reason thus in regard to 
any other profession, even to those in which original 
power tells most, and in which the mechanic is most 
easily distinguishable from the inspired artist." 

3. The study of the science of education and the 
subsequent deduction of methods by the individual 
teacher will not satisfy the demands of a common-sense 
pedagogy, because — 

(«) The average man fails to trace back deductions 
to their underlying principles for verifications, and rests 
content in the belief that his practice is in accord with 
his theory. The teacher is no exception, — he masters 
his science, then, blissfully unconscious, proceeds with 
wrong applications. 

(b) It takes years of unaided effort to build up a 
satisfactory method, and if we take into consideration 
the fact that the number of years thus spent in prepa- 
ration is greater than the average teacher devotes to 
his profession, it will be readily seen that the benefits 
accruing to the teacher in power and breadth, not to 
mention freedom, will scarcely offset the injustice to 
which the child is subjected during that period, — and 
the child has a right to object to being made the subject 
of experiment. 

4. Nor is the claim valid that training in specific 
methods unfits the teacher for spontaneous, independ- 



PREFACE. 9 

ent, characteristic work. "No natural educator," says 
W. Rein, "is so gifted through divine favor from the 
beginning as to be able to reach the highest results 
entirely without the aid of methodical schooling, and 
there will never be a method so wonderful as to be 
able to supplant the power of a strong personality. . . . 
This truth must nevertheless be advanced against the 
scorners of all method : even the most happily consti- 
tuted nature, the teacher by divine grace, is not re- 
stricted nor rendered ineffective by the directions of 
method ; on the contrary, his activity is promoted and 
insured of its effectiveness." There is indeed no more 
valid reason for the claim that the study of method 
tends to suppress the individuality of the teacher than 
there was in the claim that the teaching of penmanship 
by printed models would crush the individuality of the 
child's handwriting. There is more in human nature, 
in human character, in the human mind than what is 
put there by the pedagogue. Nay, more, nature will 
have its*say in spite of him; so that the student with 
a taste for investigation will not renounce his work 
simply because such work has been facilitated for him ; 
while he who is not so inclined will certainly not be 
assisted by the fact that the study of methods formed 
no part of his preparation. 



I. _READING. 

There are three stages in the process of learning to 
read : the first aims at knowledge ; the second, at 
power ; and the third, at fluency. Each stage calls 
into play different processes of the mind and demands 
a representative mode of treatment at the hands of 
the teacher. 

First Stage : Presentation. 

The purpose of the early lessons in reading is to 
familiarize the child with the forms of certain words. 
To this end, the teacher chooses familiar and simple 
words or sentences, pronounces them before the class, 
and writes them on the blackboard. Attention and 
repetition complete the work. 

During this stage no attempt is made at analysis ; 
words are studied as wholes, and the aim of the lesson 
is attained when the child, at the sight of the symbol, 
recalls readily the idea it stands for. 

Second Stage : Comparison. 

The word-method * is the proper vehicle for initiat- 
ing the child into reading. Symbol-recognition, how- 

1 The advantages of this method are : 

1. It associates the unknown written word with the known spoken 
word. 

2. It accords with the Culture Epoch Theory. 

3. It presents material for inductions. 

4. It appeals to the child's utilitarian tendencies. 

11 



12 READING. 

ever, is not an end in itself, but a preparation for the 
more serious and important work of inducing the 
phonic elements of our language and the powers of 
the letters. In the subsequent lessons, therefore, the 
child's attention is directed to the comparison of words 
having common characteristics. These, in obedience 
to the mind's tendency, gradually dissociate themselves 
from the words of which they form parts, and become 
generalized into definite symbol-sounds. 

As a brief illustration of the method : The child who 
has learned to recognize as symbols some such words as 

tar fat 

car fan 

far for 

will experience but little difficulty in abstracting the 
common elements and in assigning to each its appro- 
priate sound. 

Third Stage : Application. 

THE ART OF READING. 

Aim: Thought. 

Means : Symbols. 

The ultimate aim of reading is thought-getting ; but 
as this presupposes a recognition of the symbols used 
in conveying thought, the teacher must bear in mind — 

Fundamentals. 
1. That the recognition of the thought and the 
recognition of the symbols are two distinct mental 
processes, having attention as their common factor. 



READING. 15 

2. That only a definite amount of attention is avail- 
able for any given period of time ; so that the greater 
the demand is on the child's attention for symbol-recog- 
nition, the less of his attention remains to be expended 
on thought-getting. 

3. That while the aim of reading is constantly held 
before the child, the teacher should strive to give him 
a gradually increased proficiency in the recognition of 
the symbols. 

4. That the ideal condition obtains when the meaus 
are employed automatically in the attainment of the 
aim. 

Method. 

1. Reading by the child. 

By throwing the child upon his own resources, he 
will eventually gain power and self-reliance. 

Aids : 

(a) Syllabication. 

Teach the child to focus his attention upon each 
separate syllable of a word. The syllables united are 
impregnable ; divided, they yield. It is only in this 
way that words of many syllables are recognized. 
(6) Comparison. 

Most of the syllabic elements the child masters dur- 
ing the first few months of his school life. If he 
masters also the powers of the letters, he has at his 
command all the means whereby he may decipher any 
apparently strange syllable. 



16 READING. 

2. Reading by the teacher. 

The child listens to a model reading lesson, while the 
teacher takes this opportunity to clear up any vague- 
ness in the content. 

3. Thought-getting and repetition. 

(a) Questions by teacher. 

(b) Answers by pupils. 

The teacher asks a question, making it brief and 
pointed ; the child reads the answer from the book. 
An opportunity for variety is here presented. 

The exercises up to this point have been in the 
nature of a preparation ; now follow : 

4. Study by pupil, preferably in the class-room. 

5. Reading by pupil. 

Model Lessons, 
(a. Report of a lesson in a 3 A grade.) 
" The wind played many pranks. He lifted many 
strong men off their feet, and set them down in the 
market place. At length, he tired of this kind of fun." 
1. During the reading by the child, the following 
words were developed on the blackboard : 



(*) 



Known 


Step* 


Unknown 


bank 


ank 

rank 

ranks 


pranks 


car 


ar 


mar- 


let 


et 


ket 
market 



READING. 19 

2. The teacher read the selection, questioned upon 
the meaning of " pranks," and commented upon " He " 
as referring to wind. 

3. Q. Why did the wind stop having more fun ? 

A. "At length fun." 

Q. What does the first story (or sentence) tell 

you? 

A. "The wind .__ pranks." 

Q. Give an example of his pranks. 

A. "He lifted market 

place." 

4. Five minutes were given to the children for silent 
study of the whole lesson. 

5. Reading by the children. 

(b. Report of a lesson in a 5 A grade.) 

" A peasant once had a faithful horse, who had grown 
old and could not serve his master any longer ; he did 
not care, therefore, to provide him with food. So he 
said to the old horse : 4 1 really do not want you any 
more, for you are of no use to me ; but if you can prove 
your strength by bringing me a lion, I will keep you as 
long as you live. In the meantime, however, just walk 
out of my stable, and go and make yourself a home in 
the fields.'" 

1. The paragraph was read aloud by a pupil and the 
pronunciation of the following words was developed : 

(a) peasant, by comparison with pleasant. 

(6) provide, syllabicated and accent noted. 



20 READING. 

The study 1 of unfamiliar words was taken up at this 
point — a slight variation of the general method, 
(a) peasant. 

Q. What is meant by a " peasant " ? 
A. (read from vocabulary). A peasant is a coun- 
tryman. 
Q. What do we generally call a countryman who 

works in the fields ? 
A. A farmer. 

Q. Who will give one word for " peasant " ? 
A. A farmer. 
(6) provide. 

Q. Parents provide children with food. What 

does " provide " mean ? 
A. " Provide " means give, supply. 

2. Paragraph read by the teacher. 

3. Questions asked to develop the thoughts of the 
paragraph : 

Q. Who are the characters in the story ? 

A. A peasant and a horse. 

Q. Why did not the master wish to feed the horse 

any longer ? 
A. " He had grown old any 

longer." 
Q. Had the horse served his master well before 

he became old ? 
A. " A peasant once had a faithful horse." 
1 An example of correlation. 



READING. 23 

Q. What did the farmer tell the horse to do ? 
A. " Just walk out in the 

fields." 
Q. What could the horse do to get back ? 
A. "If you can prove as you 

live." 

4. Paragraph studied silently by class. 

5. Pupils called upon to read, others to reproduce 
orally. 1 

1 Correlation. 



II. — DICTATION. 

Aim : To represent thought in correct written form. 



Topics included : 






1. Spelling. 


5. 


Abbreviations. 


2. Homonyms. 


6. 


Paragraphs, Stanzas, 


3. Capitals. 


7. 


Letter Forms. 


4. Punctuation. 


8. 
SPELLING. 


Use of Dictionary. 



Aim: To represent words according to fixed forms. 

Spelling, as the art of recording sounds by means of 
symbols, has for its basis the recognition of elementary 
sounds and their representative signs or letters. In a 
phonic language, analysis and comparison suffice for 
the study of its spelling ; in the English language, how- 
ever, its spelling, though based on rational grounds, 
presents so many anomalies, that the memory must be 
called upon to assist the other two processes. 

Fundamentals. 

1. Analysis is the basic process in spelling — the 
human mind must first recognize and hold in attention 
the syllable before it can determine the elements of 
which it is composed and the symbols which represent 
them. 



DICTATION. 27 

2. The analytico-synthetic method should by em- 
ployed in spelling — word, syllable, letter, word. 

3. Each spelling word, when first presented for 
study, should be introduced in a sentence. 

4. Each spelling lesson should include both an oral 
and a written exercise — the former for study, the 
latter for testing. 

5. Rules for spelling should be learned inductively 
and applied deductively. 

6. The greater the number of the association links 

as the result of the activity of the various senses, the 

readier the recall. 

Method. 
Oral Spelling : 

1. The word is first pronounced, then written on the 
blackboard, syllabicated by vertical lines, and its accents 
marked. 

2. Individual children are then called upon to spell 
as follows : 

(a) The word pronounced. 
(6) The first syllable pronounced, 
(e) The first syllable spelled. 
The remaining syllables having been treated like- 
wise, there follows : 

(<i) The word pronounced. 

3. The children with arms extended in front of 
them, and with fingers pointing to the word, write it 
several times in the air, while the teacher signals each 
new letter by counting. 



28 DICTATION. 

4. The same exercise with eyes closed. 
Written Exercise : 

1. Words dictated. 

2. Words spelled, one pupil at the same time writing 
them on the blackboard. 

3. Errors noted by drawing a line through the word. 

4. Papers collected and returned. 

5. Corrected forms written above the errors and 
below the exercise. 

6. Study and recitation of corrections. 

General Remarks. 

1. A dictation should accompany each spelling exer- 
cise. 

2. Each day's spelling lesson should include the new 
words of the previous day. 

3. The spelling of words whose meanings are not 
known is a waste of time and energy. 

HOMONYMS. 

Method. 

1. Teach through the medium of the sentence. 

2. Teach one word of a group at a time. 

3. When the words of a group have been mastered 
separately, they should then be compared. 

4. Test: 

(a) Write sentences on blackboard, omitting the 
homonyms, and require pupils to supply 
them. 



DICTATION. 31 

(5) Require pupils to give sentences with certain 

homonyms. 
(<?) Give sentences orally, and require children 

to spell the homonyms. 

CAPITALS. 

Steps in Method. 1 

1. Recognition in reading matter. 

2. Mastery of rule. 

3. Application. 

PUNCTUATION. 

Punctuation is a device for the ready interpreta- 
tion of written languages ; hence, ability to punctuate 
demands both a rational and a formal process — the 
former corresponding to the mental process which de- 
termines the separate groups of words, and the latter 
dealing with the kinds of punctuation marks that 
should be inserted. 

PARAGRAPHING. 

Paragraphing, like punctuation, is a device which 
has a mechanical and a rational side — the former deal- 
ing with its printed form, and the latter with the rela- 
tion which its component sentences bear to itself. 

Fundamentals. 

1. Necessity for punctuation and for paragraphing is 
determined by the content of the selections ; hence, in 
1 Same for abbreviations and letter forms. 



32 DICTATION. 

teaching, emphasis should first be laid on the content; 
then on the form. 

2. Since mastery of the thought is a prerequisite to 
punctuation and to paragraphing, the selections chosen 
should be adapted to the understanding of the pupils. 

3. Punctuation and paragraphing should be learned 
inductively and applied deductively. 

Method. 
The Period: 

1. John is a good hoy. 

2. I have a new book. 

3. The tree has many leaves. 

etc., etc., etc. 

The child notices that the first sentence ends in a . 

He notices that the second sentence ends in a . 

He notices the . at the end of each subsequent sentence. 

He concludes 1 that each similar sentence must end 

in a . 

_ THE PARAGRAPH. 

Form : 

This knowledge the child obtains by having his atten- 
tion called to the indention which marks the beginning 
of each paragraph. 

Content : 

Successive paragraphs are read and analyzed into 
their component sentences. Each group is then shown 
to deal with some particular topic of the theme. 

1 By a similar reasoning he concludes that each sentence must 
begin with a capital. 



DICTATION. 35 

Example : 

kt Once there was a little girl who had a beautiful red 
hood. It was as red as the sun when it sets behind the 
clouds in summer. Her grandma gave it to her. 

" ' This is a fairy hood,' said her grandma. ' It will 
always look just as pretty as it does now. You must 
wear it every time you go out.' " 

Analysis : 

The topics. 

1. The Hood. 

1. (a) The owner of the hood. 
(£>) The color of the hood. 

(<?) How the owner obtained the hood. 

2. What Grandma Said. 

2. (a) Of the peculiar attribute of the hood. 

(b) Of the peculiar virtue of the hood. 

(<?) In the shape of advice concerning the hood. 
Application : 

(a) To reading matter. 

(6) To writing from topical outlines. 

(c) To arrangement of sentences into topics. 
(b and e are correlated with composition.) 

_ USE OF DICTIONARY. 

Presupposes : 

1. A knowledge of the letters of the alphabet and 

the order of their sequence. 

2. Ability to determine the proper word by a refer- 

ence to the content. 



36 DICTATION. 

Dictation as a Test. 
(The Fifth Herbartian Step.) 

1. Row the Exercise is Conducted: 

(a) At least one child writes on the blackboard, 
while the rest write on paper. 

(5) The number of words dictated consecutively 
depends upon the mental capacity of the 
child and upon the nature of the selection. 

(e?) Only groups of related words are dictated. 

(d) The speed of the dictation is limited by the 

ability of the majority of the class to take 
down the exercise verbatim. 

(e) No repetition is permissible unless it is de- 

manded by the content of the selection. 

2. Method of Correction : 

(a) Individual children are called upon to read 
their written exercises, the rest meanwhile 
inserting omissions. 

(7>) The blackboard exercise as " proof-sheet," 
the attention is directed first to the para- 
graphs as wholes, then to the separate sen- 
tences as wholes, and lastly to the elements 
of each sentence. 

(c) Classes that are supplied with dictionaries 
should use these to look up the correction 
of misspelled words and the meanings of 
unknown words. 



DICTATION. 39 

(t?) Correction is made by passing a line through 
the error, rewriting correctly above the 
word, and in the case of a misspelled 
word, writing it also below the exercise. 

General Directions. 

(a) Time must be assigned for the study of the 
corrected forms, and for recitation by sev- 
eral pupils. 
(5) Each sheet of paper should contain two origi- 
nal dictations; if redictation is deemed 
advisable, it should be taken on a separate 
sheet. 
(V) An oral review of exercises should be had at 
least once a month, when children are given 
an opportunity to restudy previous correc- 
tions. A recitation should follow. 

Model Lesson. 1 
(Report of a lesson in a 6 A grade.) 

The teacher dictated the following in groups of 
words included between the vertical lines : 

As the fair happened on the following day, | I had 
intentions of going myself. | 

" No, my dear," said my wife, | " our son Moses is a 
discreet boy, | and can buy and sell to very good ad- 
vantage." | 

1 The main aim of the lesson was to test the children's knowledge 
of paragraphs and quotation marks. 



40 DICTATION. 

As I also had a good opinion of my son's pru- 
dence, | I was willing enough to trust the business 
with him. | 

Method of correcting the "Proof-sheet." 

Teacher. How many paragraphs in this dicta- 
tion ? 

Pupil. Three. 

T. Why ? 

P. Assigns reason. 

T. How many complete thoughts in the first para- 
graph ? 

P. One. 

T. In the second ? 

P. One. 

T. In the third ? 

P. One. 

T. Do you notice any mistakes in the use of the 
quotation marks? In the use of any other punctua- 
tion marks ? 

P. Assigns reasons for corrections. 

T. Do you notice any other mistakes in the first 
sentence ? in the second sentence ? in the third sen- 
tence ? 

P. Makes corrections. 

Each child next corrected his own exercise, using 
the blackboard as guide, and studied the correct forms 
for a subsequent recitation. 



DICTATION. 43 

The "Proof-sheet." 
As the fair happened on the following day, I had 

intentions *T[ i> 

intensions of going myself. " No, my dear, said my 

a discreet j. 

wife, our son Moses is a discret boy and can buy and 

advantage 

sell to very good advantag." prudence 

As I also had a good opinion of my son's prudense, 
I was willing enough to trust the business with him. 



1. intentions 


5. discreet 


2. U 

3. " 

4. " 


6. , 

7. advantage 

8. prudence 



HI. — COMPOSITION. 

1. The arrangement of thoughts in their logical order, 
and 

2. Their expression in correct language. 

A child has neither the information, the reasoning 
power, nor the training to write a masterly treatise ; 
but under proper conditions he can be furnished with 
the information ; he can be taught to express it gram- 
matically ; and he can be trained to arrange his thoughts 
logically. It must be borne in mind, however, that the 
development of a higher faculty through exercise pre- 
supposes a particular exercise of a lower faculty. 

Fundamentals. 

1. The subject-matter and the manner of treatment 
should be adapted to the child's interests and capacities. 

2. Oral composition should be practiced long before 
an attempt is made at written composition — the pen- 
manship and the spelling are obstacles. 

3. Composition should not be subordinated to other 
studies, but should be made an aim in itself. 

4. The child's self-activity should be called into play 
at every opportunity. 

44 



COMPOSITION. 47 

Kinds of Composition : 

1. Reproduction of a model (perception). 

2. Imitation of a model (imagination). 

3. Writing from topical outlines (judgment). 

4. Wholly original (reasoning). 

Method. 

Presentation : 

1. 2. 3. (a) Models read by teacher and dis- 
cussed (or subject and treatment discussed for 2 
and 3). 

(6) Orally reproduced by several children. 

(<?) Compositions written, and several read. 

4. (a) Title written on blackboard. 

(6) Children volunteer information, and write 
corrected forms on the blackboard. 1 

(c) Attention directed to sequence of sentences 
and to their division into paragraphs, giv- 
iug to each of the latter its appropriate 
heading. 

(c?) Only headings left on blackboard, and chil- 
dren called upon to give oral expression 
to the thoughts. 

(e) Compositions written and several read. 

1 This order of presentation is changed as soon as the ability of the 
children warrants it, by choosing as the initiatory exercise the deter- 
mination of the topical outline and the volunteering of information 
by the children. 



48 COMPOSITION. 

Correction : 

1. Compositions written on blackboard. 

2. Criticised by class (see Dictation, Method of Cor- 
rection, b and d). 

3. Compositions rewritten. 

General Remarks : 

1. Make haste slowly — quality rather than quantity. 

2. The oral work is the most important part of the 
exercise. 

3. Compositions are written on blackboard before 
or after the regular school session. 

4. Home lessons may be assigned in the upper grades. 

5. Other compositions besides those written on the 
blackboard may be criticised, but always in the pres- 
ence of the child. 

Model Lessons. 
a. Reproduction : 

(Report of a lesson in a 5B grade.) 

1. The teacher read the following fable to the class : 

THE DOG IN THE MANGER. 

There was once a dog who lay all day long in a 
manger where there was plenty of hay. A horse, a 
cow, a sheep, and a goat came one by one and wanted 
to eat the hay. The dog growled at them and would 
not let them have so much as a mouthful. Then an 



COMPOSITION. 51 

ox came and looked in, but the dog growled at him 
also. 

" You selfish fellow ! " said the ox, " you cannot eat 
the hay. Why do you want to keep it all to your- 
self ? " 

2. Several pupils were now called upon to repro- 
duce the reading orally and, at the conclusion of each 
recitation, the class made the necessary corrections. 1 

3. Compositions were written. 

4. After a number of pupils had been called upon to 
read their compositions, which served both as models 
and as subjects for criticism, the attention of the class 
was directed to the correction of the " proof-sheet." 

5. Compositions that had been criticised either by 
the class or by the teacher were rewritten. 2 

(Example of a Child's Reproduction.) 

THE DOG IN THE MANGER. 

There was once a dog who lay all day long in a 
manger which was full of hay. Soon a horse, a cow, a 
sheep, and a goat came one by one to eat the hay, but 
the dog growled at them and would not let them have 
even a mouthful. 

Next, an ox came to eat the hay and the dog growled 

1 In this instance, also, the teacher, by several well-directed ques- 
tions, led the children to recognize the number of paragraphs into 
which the reading was divided — another example of correlation. 

2 The teacher himself corrected about one third of the composi- 
tions of the class, in the presence of individual pupils. 



52 COMPOSITION. 

at him too, but the ox said, "You foolish fellow, you 
cannot eat hay. Why don't you let somebody else 
have it then?" 

b. Imitation: 

(Report of a lesson in a 6B grade.) 

1. The teacher read the following to the class : 

THE CAT, THE MONKEY, AND THE CHESTNUTS. 

A Cat and a Monkey were sitting one day by the 
hearth, watching some chestnuts which their master 
had laid down to roast. The chestnuts had begun to 
burst with the heat, and the Monkey said to the Cat : 

" It is plain that your paws were made to pull out 
those chestnuts. Your paws are, indeed, exactly like 
our master's hands." 

The Cat was greatly flattered by this speech and 
reached forward for the tempting chestnuts; but 
scarcely had she touched the hot ashes than she drew 
back with a cry, for she had burned her paw. She 
tried again, and made out to get one chestnut; then 
she pulled another, and a third, though each time she 
singed the hair on her paws. 

When she could pull no more, she turned, and found 
that the monkey had taken this time to crack the chest- 
nuts and eat them. 

2. Several compositions were read, discussed, and 
criticised. 

3. 4, 5, same as in a. 



COMPOSITION. 55 

(Example of a Child's Imitation.) 

THE GIRL, THE BOY, AND THE CANDY. 

One day, while a little girl was sitting under a tree 
eating some candy, a boy came up to her and asked her 
to get him some water from a spring near by. " Why 
don't you go yourself? " she asked. 

" Because it will taste better if you'll get it for me," 
he answered. This speech so nattered the little girl 
that she laid her candy on the grass and ran off to fetch 
the water. 

But, alas ! when she returned, there was neither boy 
nor candy to be seen. That day she learned a lesson 
she never forgot. It is : " Never allow people to natter 
you into doing things which they can do for them- 
selves." 

c. Topical outline : 

(Report of a lesson in a 7 A grade.) 

ROBERT FULTON. 

1. A talk on the life of Robert Fulton by the teacher, 
and an oral reproduction by the children. 

2. The following topics were written on the black- 
board by the teacher : 

(a) Robert Fulton at home and at school. 
(5) A youthful invention. 

(e) His greatest and most successful invention. 
(d) His death. 

3. 4, and 5 same as in a. 



56 COMPOSITION. 

(Example of a Child's Composition.) 

ROBERT FULTON. 

Robert Fulton was born in 1765, in Little Britain, 
Pa. When he was three years old, his father died, and 
his mother was left to take care of his education. She 
taught him until he was eight years old. He did not 
have a great liking for books. He was very much 
interested in mechanical drawing. 

Robert came late to school one day. After being 
reproved by the schoolmaster, he said, " I was at a shop, 
hammering a piece of lead into a pencil." Soon after, 
the children were using the same kind of pencils. 

Fulton invented the first steamboat. He called it 
the Clermont, after his friend Livingston's home on the 
Hudson. Its first trip took place in 1807. It started 
from New York and sailed up the Hudson River to 
Albany, which was a distance of one hundred and fifty 
miles. It took her thirty-two hours to make this trip. 
People stood along the banks of the Hudson River, say- 
ing they didn't think the boat would go. They called 
it "Fulton's Folly." They were very much surprised 
when they saw the boat begin to move. 

Robert Fulton died in 1815. The world owes a 
great deal to this man. 

d. Original : 

(Report of a lesson in an 8 A grade.) 
1. The pupils, under the supervision of the teacher, 



COMPOSITION. 59 

determined upon the following headings, without in- 
sisting upon the order of their treatment : 

(a) What food does. 

(J) Kinds. 

(<?) Preparation. 

(d) Necessity. 

(e) Dangers. 

2. The pupils volunteered a number of sentences, 
each of which, in turn, was first examined for its syn- 
tax, and, if approved, was written on the blackboard 
and examined for written mistakes. It was then 
assigned to its proper heading. 

Sentences Volunteered. 

(a) Food builds up the body. 

Our bodies are constantly wearing away, 
therefore we must have food to repair 
them. 
Pure food gives pure blood. 
(3) There are three kinds of food, animal, vege- 
table, and mineral. 
Air is a food. 

Some parts of the body need special food, as, 
for instance, the brain needs phosphorus, 
and the bones need lime, 
(c) All food should be well chewed before it is 
swallowed, or it will cause indigestion. 
We cook food to help digestion. 



60 COMPOSITION. 

(d) If you do not get enough food, your body 

will become weakened. 

We must eat pure food in order to be healthy. 

Man cannot live on one kind of food, but must 
have several kinds mixed in right propor- 
tion. 

(e) There are some things we must avoid, such as 

tobacco, drugs, and liquor, for they are 

poisonous. 
Liquor is dangerous to the body because it 

contains alcohol. 
We must exercise the muscles in order to get 

food from the blood. 1 

3. Everything but the headings was erased from the 
blackboard; and after several pupils had reproduced 
the subject orally, the class wrote the composition. 

4 and 5 same as in a. 

1 It need surprise no one to learn that the pupils did not give the 
sentences in the exact order given above. However, they made a 
laudable attempt to confine their attention to a single heading at a 
time — a practice which, if repeated sufficiently often, will tend to form 
a habit of concentrated thinking in composition writing. 



IV. _ GRAMMAR. 

Aims : 

1. Intellectual discipline. 

2. ^Knowledge. 

Grammar deals with certain phases of the content 
and the form of the sentence. It treats (a) of the 
relations and the functions of the thought-elements, and 
(5) it presents certain forms of expression which usage 
has imposed upon our language. The standard of the 
one is in the mind, hence universal and unchangeable ; 
the standard of the other is in custom, and hence sub- 
ject to change. Analysis, which concerns itself with 
content, belongs to the former ; syntax belongs to the 
latter. Briefly stated, grammar on its rational side is 
the psychology of the sentence ; on its formal side it is 
a guide to correct usage. 

Fundamentals. 

1. The sentence is the basis of the study of gram- 
mar. 

2. The analysis of a sentence into its elements pre- 
supposes : 

63 



64 GRAMMAR. 

(a) A clear understanding of the thought con- 
veyed. 
(6) A sufficient degree of mental development. 

3. Like all studies that yield to scientific treatment, 
grammar should be taught inductively and applied 
deductively. 

Method. 

Aim : The concept, Sentence. 

Place on blackboard a series of unrelated words ; as, 
this, told, birds, farm, a, lived, story, fly, Franklin, boy, 
on ; then direct your pupils' attention to a comparison 
of these with their rearrangement into the combina- 
tions : birds fly, the boy lived on a farm, Franklin told 
this story, etc. 

Generalization (naming) and application should fol- 
low in this as in subsequent lessons. 

Aim : Analysis of the Simple Sentence. 



1. 


The sun 


shines. 


2. 


The boy 


lived on a farm. 


3. 


Franklin 


told this story. 


4. 


The city of Boston 


is the capital of Massachu- 
setts. 



Exercise | must be taken daily. 



GRAMMAR. 



07 



By means of questions lead your pupils to recognize 
the two coordinate elements of each sentence, as fol- 
lows : 



Questions by teacher. 

1. What shines? 

What does the sun do? 
What is said concern- 
ing the sun? 

2. Who lived on a farm ? 
What about the boy ? 

3. Who told this story? 
What did Franklin do, 

or what does this sen- 
tence tell you con- 
cerning Franklin? 

4. What is the capital of 

Massachusetts ? 
What is said of the city 
of Boston? 

5. What must be taken 

daily ? 
What concerning exer- 
cise? 



Answers by pupils. 
The sun 



shines. 

The boy 

lived on a farm. 



Franklin 



told this story. 
The city of Boston 

is the capital of Massa- 
chusetts. 

Exercise 



must be taken daily, 



By comparison, the idea is now brought prominently 
to the pupils' attention that the sentence concerns itself 
with something of which the rest of the sentence treats. 



68 GRAMMAR. 

That " something " is the subject ; what is stated con- 
cerning the subject is the predicate. 1 

Aim : The concept, Complex Sentence. 

1. A. I was thus employed. 

B. The enemy discharged several thousand ar- 
rows. 
We have here two simple sentences which give no 
evidence of connection or of relationship. 

C. While I was thus employed, the enemy discharged 
several thousand arrows. 

By skillful questioning, the pupil can be led to per- 
ceive : 

1. That the two statements have been incorporated 
into one (clauses). 

2. That the two statements are now not only con- 
nected but related — A being the occasion of B (de- 
pendence). 

3. That A relates to "discharged" (adverbial). 
These ideas must now be mastered ; and this is done 

by (a) repetition, (b) type-sentence. 

The tjrpe-sentence is of value both as the repository 

1 The difficulty children encounter in discovering the elements of a 
sentence is due largely to the neglect to emphasize the dependence of 
these elements upon each other. To obviate this difficulty, the child 
should be trained to think of the subject-predicate, and of predicate- 
subject and to recite the subject and predicate together, laying empha- 
sis upon the particular element called for in the answer. Thus in the 
sentence, "Birds fly," if the subject is asked for, the child recites, 
"Birds [fly] " ; if the predicate, " [Birds] fly." 



GRAMMAR. 71 

of the ideas and as the concrete standard with which to 
compare similar sentences. Hence, every new concept 
should have a type as its accompaniment. 

Repetition is essential to stamp knowledge upon the 
mind — not the kind of repetition that degenerates into 
a mechanical process, but the one that calls to its assist- 
ance variety, interest, and thought. In conformity 
with this, the following method is suggested for the 
study of the type-sentence : Having induced the prin- 
ciple, the pupils, while applying it to a variety of sen- 
tences, should be called upon at frequent intervals to 
apply it to the type-sentence, until they are able to 
recall readily the sentence and the process. 

Model lesson to determine the structural character of a 
sentence. 

To determine the structural character of a sentence, 
we need to know : 

1. The number of clauses composing it. 

2. The nature of the relationship existing between 
these clauses. 

Model. 

What kind of a sentence is the following ? 

"Accordingly, when he came to the convention in 
Independence Hall as a delegate from Virginia, he 
chose a good seat where he could hear all that was 
said." 

First Step: to determine the number of clauses. 



72 GRAMMAR, 

1. (Accordingly) he chose seat. 

2. (when) he came Virginia. 

3. (where) he could hear all. 

4. (that) was said. 

The idea of the simple sentence is now eliminated. 

Second Step: to determine the dependence or inde- 
pendence of the clauses. 

1. independent clause. 

2. relates to " chose " ; dependent clause. 

3. relates to " chose " ; dependent clause. 

4. relates to " all " ; dependent clause. 

Third Step : Conclusion : Complex Sentence. 
The Diagram. 

As a graphic representation, it is of great value, espe- 
cially for beginners, and should be employed for both 
analysis and synthesis of each sentence. 

The Parts of Speech. 

Should be taught inductively through the medium 
of the sentence, with the Formal Steps for guides. 

Parsing. 
Process: deduction. 

Warning: do not invert the order of the deductive 

process — major premise, minor premise, conclusion. 

Examples of parsing : 
John bought a new book. 
1. Parse "new." 



GRAMMAR. 75 

Major premise (adjectives are words which modify 
nouns or pronouns). Children are credited with this 
knowledge, and are not called upon to repeat it. 

Minor premise: " new " describes " book." 

Conclusion: "new" is an adjective. 

Briefly: " new " describes " book " and is an adjective. 

2. Parse "John." 

Proper noun subject of "bought," and 

is in the nominative case. 

False Syntax. 

The recitations and the compositions of the pupils 
should furnish material for the correction of sentences. 



V. — LITERATURE. 

A masterpiece of literature may be studied for com- 
prehension, or read for appreciation, or analyzed for 
criticism. 1 This must be borne in mind in discussing 
the subject of method in literature. 

Fundamentals. 

1. As the aim determines the mental processes en- 
gaged in its attainment, it also determines the method. 

2. As comprehension must precede appreciation, 
and appreciation must precede criticism, the mental 
processes engaged in executing these aims bear to 
each other the relation of the simple to the complex. 
Hence the method for appreciation includes that for 
comprehension, and the method for criticism includes 
that for appreciation. 2 

1 Criticism is beyond the province of the elementary schools, and 
will not be discussed here. 

2 The opinion expressed by certain optimistic theorists that a child 
can appreciate literature that he does not comprehend is about as 
sound as would be the claim that he can criticise a selection he does 
not appreciate. As a matter of fact, the child who finds pleasure in 
listening to the reading of a passage he does not understand does so, 
not because he appreciates the literature as such, but because he is 
attracted by its delivery. This view is not poetic, but neither are a 
great many other truths. 

76 



LITERATURE. 79 

3. Comprehension presupposes a knowledge of the 
intellectual elements of the composition, and of their 
relations. 

4. Appreciation demands, as conditions to success: 

(a) Intellectual mastery of the exercise. 

(5) Feelings sufficiently developed to apperceive 

the sentiments conveyed in the selection. 
(<?) The imagination awakened to the proper 

degree of activity. 
(cT) An attractive presentation. 

Conclusion: As appreciation is the aim of this study 
in the elementary schools, a masterpiece will call for 
either one reading or two. If the selection is one 
which presents no difficulty to the understanding of 
the child, one reading, and that for appreciation, is all 
that is essential ; otherwise, two readings will be found 
necessary — the first for intellectual mastery, and the 
second for appreciation. 

Method. 
Preparation. This will include : 

(a) The name of the author and as much of his 
life's history as is warranted by the selection. Culture 
demands this. 

(5) The occasion of the masterpiece. This gives the 
proper background and atmosphere to the thoughts and 
sentiments of the author, and paves the way to a proper 
apperception. For example, in order that a child may 



80 LITERATURE. 

appreciate the reading of "Old Ironsides," he must first 
become acquainted with the motive which actuated the 
author to write this stirring appeal. 

Presentation. 

(a) Comprehension. When a pupil has read through 
a paragraph carefully and has failed to grasp its mean- 
ing, he should search out the cause or causes of his fail- 
ure and' proceed to remove them by a study of : 

1. The meanings of unknown words. The individ- 
ual members of the class, the dictionary, the teacher, 
and, on rarer occasions, the content, may each become 
the source of information. 

2. Figures of speech. These variations of the ordi- 
nary mode of expression serve a most important pur- 
pose in literature. They add grace and refinement to 
thought ; they clarify it ; they invest it with new 
charms ; they endow it with greater vigor and power. 
They therefore demand a proper share of the child's 
attention as living and vital forces of our language. 

3. Allusions. These should be regarded in the same 
light as are unknown words whose meanings are sought. 
That is, their study should not go beyond what is neces- 
sary for a clear comprehension of them. To do more 
than this tends to introduce breaks in the continuity of 
the lesson. Brevity is the soul of an allusion. 

4. The grammatical construction. The words, 
phrases, and clauses of each sentence must be grasped 
in their true relations before the exact meaning of 



LITERATURE. 83 



the author can be known. Hence, grammar is called 
upon at this stage to complete and to insure thorough 
comprehension. The extent to which grammar should 
be utilized in the study of a masterpiece of literature 
must be measured by the need of it for a clearer under- 
standing of any portion of the selection. Its use must 
be limited by the consideration that it is of value here 
only in so far as it aids the child to discover and to 
strengthen the weak points in his knowledge. 1 

(6) Appreciation. 2 While the dictionary, the rheto- 
ric, the reference book, and the grammar, when com- 
bined, present a complete array of means for mastering 
any suitable selection, they cannot, unaided, awaken and 
develop appreciation. This can be brought about only 
by the teacher who is gifted with appreciation, power, 
and insight. Here method concerns itself less with 
specific directions for the conduct of the lesson, than 
with the inquiry into the qualifications which a teacher 

i The question of an immediate rereading of each paragraph that 
has been studied, and of the final reading of the whole selection for a 
general intellectual survey, can be answered only by the teacher, who 
is guided by the varying conditions which confront him. 

2 The appreciative reading of a masterpiece need not follow imme- 
diately upon the completion of its study. Indeed, the opinion is ventured 
that better results will follow if several months are allowed to intervene 
before entering upon the appreciative reading of a selection that had 
previously been read for comprehension. In this connection, the plan 
is suggested of assigning to each grade two classes of masterpieces, — 
one, easily comprehended, for appreciative reading ; the other, of oppo- 
site character, for thorough study. The latter may then be read for 
appreciation during some subsequent term. 



84 LITERATURE. 

must possess, or must strive to acquire, if possible, in 
order that he may inculcate in the child a desire for 
good literature. 

A teacher who is himself a lover of good literature 
cannot fail to communicate more or less of this feel- 
ing to his pupils ; for, " Love beams from the eye, 
glides from the tongue, and finds its echo in some 
heart." 

A teacher who combines oratorical powers with his 
love for literature not only awakens and maintains 
the interest of the pupils in any particular selection, 
but also instills in them a desire to become acquainted 
with other selections. 

And, finally, if, in addition to his other qualifications, 
the teacher possesses teaching tact, — if he knows how 
to adapt his instruction to the child's apperception, in- 
terest, and imagination, — he can best attain the aim 
of this study by merely reading the selection to the 
class, and by insinuating information, comment, or pre- 
cept, — in short, whatever his native or acquired tact 
may suggest. 



VI. — NATURE STUDY. 

Aims : 

1. To inculcate a feeling of sympathy for nature. 

2. To learn nature's laws and their application. 

This study is a composite of a number of sciences 
which fall under the two general divisions of observa- 
tional and experimental science. To the former belong 
botany, geology, zoology, meteorology, and astronomy ; 
to the latter belong physics and chemistry. 

Early man spent a large part of his time in studying 
the observational sciences. He was interested in the 
plants, in the soil, in the animals, in the weather, and 
in the climate, for a very obvious reason — the knowl- 
edge was essential to the maintenance of his life. As 
he advanced in civilization, he devoted a gradually in- 
creased share of his attention to physics and chemistry, 
no longer for the purpose of preserving his life, but to 
economize effort. 

The culture epoch theory, if followed implicitly, 
would necessitate the study of all the material which 
interested man in his natural state, without taking into 
account the changed conditions which confront the 
student of to-day. This is a fallacy into which the 
overzealous advocates of nature study have fallen. If 

87 



88 NATURE STUDY. 

a city child, because of his innate love for the beautiful, 
is interested in flowers, does it follow that he will be 
interested in the snake simply because some forefather 
of his in the remote ages needed that knowledge for his 
personal safety ? If natural phenomena appeal to the 
child, will the knowledge of cecropia, polyphemus, or 
cynthia have the same effect ? What possible reason, 
utilitarian, intellectual, aesthetic, or ethical, is there for 
burdening the child's mind with knowledge of spiders, 
slugs, toads, and bats ? 

A little of everything and not much of anything is 
the motto carried out to an extent little dreamed of 
even by Comenius. Not only is this study made to 
include seven sciences, not to mention the sciences 
which are directly or indirectly, logical^ or psycho- 
logically, sensibly or nonsensically, correlated with it, 
but each science is loaded down with an overwhelming 
amount of material, some of which is uninteresting to 
the child, some is unessential, and some is suitable only 
for the specialist. If observational science is confined 
to that portion of earth study and of natural phenomena 
that is essentially a preparation to the study of geogra- 
phy, and if a little botany is added to it, the child has 
all he can cover without skimming during the first few 
years of his school course. 

Fundamentals. 

1. Observational science should precede the experi- 
mental science. 



NATURE STUDY. • 91 

2. Observation should be regarded as a preliminary 
step to the conception of the causal idea. 

3. Basic considerations in the choice of material : 
the useful, the beautiful, the true. 

4. Observation, as well as experiment, should be 
directed, and preferably, to material with which the 
child is already familiar. 

5. Observation and experiment should be accurate 
in order to lead to good intellectual habits. 

Method. 

Observation should place the child in direct contact 
with the object or phenomenon to be observed. A child 
can get a clear idea of a plant only by seeing, touching, 
smelling, and perhaps tasting it. A picture gives him 
only its form and color, while a verbal description gives 
him — a lot of words. Again, a prince of India denied 
the existence of water in solid form, and no amount of 
persuasion convinced him. He needed to be brought 
face to face with the phenomenon. 

From observation the child passes to experiment, 
which is the more complex process and includes the 
former. Here the teacher must be guided by the 
maxim to have the child make his own inductions and 
deductions. 

Example of induction ; 

The child is given a ruler, a fulcrum, and several 
equal weights. He places a weight on each end of the 



92 . NATURE STUDY. 

ruler, which he balances on the fulcrum. He moves 
one weight nearer the fulcrum and he finds that he 
must increase the power-distance or the weight in order 
to preserve the balance. After several observations of 
like character, he concludes that P x Pd= Wx Wd, a 
discovery which he proceeds to apply. 



VII. — GEOGRAPHY. 

Aims: 

1. Reverence. 

2. Liberal-mindedness. 

A characteristic of modern education is that it aims 
at the harmonious development of the threefold nature 
of man — physical, mental, and moral. Another of its 
characteristics is that, while it conceives it possible to 
exercise the mental powers through the study of one 
group of subjects, it insists that certain studies are 
inherently fitted for particular mental processes. In 
accordance with this decision, it has assigned nature 
study to the elementary school primarily for observa- 
tion, reading for imagination, arithmetic and grammar 
for reasoning, history for prudential morality, and 
geography for its highest aim, reverence and liberal- 
mindedness. Geography, properly taught, yields the 
material which, if elaborated in accordance with the 
mind's laws, make possible the attainment of these 

aims. 

Fundamentals. 1 

I. The conditions to reverence are : 

1. (&) A knowledge of the earth as the dwelling- 
place of man (utilitarian). 
(5) A knowledge of the natural forces that 
are at work upon it (utilitarian). 

1 Read, in this connection, Fundamentals under History. 
95 



96 GEOGRAPHY. 

2. A knowledge of the relation between the earth 

and the natural forces (rational). 

3. (V) Contemplation of the earth beautiful (aes- 

thetic), and 
(5) Contemplation of causes and effects, lead- 
ing up to the conception of the Uni- 
versal Cause (ethical). 

II. The conditions to liberal-mindedness are : 

1. (a) A knowledge of the physical characteris- 

tics of other lands. 
(b) A knowledge of their inhabitants. 

2. A knowledge of the influence that habitation 

has upon man. 

3. An appreciation of the truth that we are 

largely the product of our environ- 
ments. 

General Considerations. 

1. The map represents the world which the child 
travels in imagination. Before he can avail himself of 
its services, however, he must master : 

(a) The elementary notions of geography. 
(5) The manner of representing and of interpret- 
ing these notions. 

2. Elementary geographical notions should result 
from observation or experience. 

3. Induction is the proper method for the teaching 
of geography. 



GEOGRAPHY. 99 

4. Ability to * interpret the map rational^ is of 
greater importance to the child than the memorizing of 
the numerous facts it contains. 

5. After the child has mastered the elementary no- 
tions of geography, it matters little to him whether the 
first topic presented for study is the whole earth or the 
city or state in which he resides, because, in reality, 
they differ in nothing except in size — a concept which 
a child can grasp or disregard as readily as an adult in 
map study. What is of importance to him, however, is 
that the topic is presented to him in bold outlines and 
subsequently filled in. 

Method. 
Elementary notions : 

Where direct observation of surface conditions is im- 
possible, clay or sand modeling should be resorted to. 
A child who has never seen a lake can get a pretty 
clear idea of it by this means, especially if the lesson is 
supplemented by pictures and by a comparison with 
some known object for the purpose of learning its size. 

Map interpretation : 

Have children draw a large object, as a blackboard, 
and call their attention to the scale ; next, draw an 
object to an exact scale. Represent objects in the room 
on the plane ; later, represent a river, a mountain, etc., 
from a clay model. Teach the cardinal points by 
means of the sun, and fix them on paper. Ask for 
exact location of any point; then teach latitude and 
longitude. 



100 GEOGRAPHY. 

Causal relations : 

Geography is not a compendium of isolated facts con- 
cerning our earth, but it is a study which regards its 
facts and phenomena as links in a long chain of causes 
and effects. As an example, the child, starting with 
location and surface as primary geographical facts, is 
led to perceive that they are the chief causes of cli- 
mate ; that location and climate influence the life of 
plant and animal, which in turn influence man's occu- 
pation, and these in turn affect commerce, travel, and 
communication. 

Indeed, there is no geographical fact which will not 
yield to the same treatment. The general surface of a 
continent ; its mountains, rivers, etc. ; its climate ; its 
population ; the size and location of its cities ; the char- 
acteristics and governments of its people ; its occupa- 
tions and productions ; its commerce, — each of these 
topics can be and should be treated in its causal relation. 

Ethical : 

Only he who feels beauty, sympathy, and reverence, 
and approaches the child with gentle tact, can awaken 
the aesthetic feelings through the contemplation of 
nature or of nature's offspring — rhythmic language ; 
can arouse sympathy through the broadening of the 
intellect ; and can transform rational insight into rev- 
erence. 



VIII. — HISTORY. 

Aims : 

1. Intellectual. 

2. Ethical. 

History is the biography of man. It not only records 
events which have a bearing upon his social life, but 
searches out their causes and effects. Such a concep- 
tion of the province of history is certainly beyond the 
mental grasp of the young child; hence, the unfold- 
ing of the subject must be made to correspond to the 
development of his mental faculties. 

Fundamentals. 1 

I. There are three stages in the development of the 
historical sense in children : 
1. Narrative. 

(a) Biography. 

(b) Events. 

(a) Sound practice demands the recognition that chil- 
dren's interests are most readily awakened by concrete 
examples. Hence, biography is chosen as the intro- 
duction to narrative which has for its basis events. 

1 Read also Fundamentals under Geography. 
103 



104 HISTORY. 

(5) The transition to events is natural, having been 
accomplished in accordance with the laws governing 
the acquisition of knowledge ; while the new material 
for study takes into account the child's increased men- 
tal development, particularly his imagination. 

2. Causal. 

(«) Events. 
(5) Institutions. 

(a) Perception of causality is a more difficult process 
than perception of events, and requires a higher degree 
of mental development. The fall of Constantinople 
and the discovery of America are historical events which 
any ordinary child can grasp ; but to trace out the rela- 
tionship of cause and effect between them is well cal- 
culated to exercise, not merely his imagination, but his 
judgment and his reason as well. 

(5) Institutions, the monuments of the world's prog- 
ress, are a fitting climax in the study of history. But 
whether they should be dealt with as mere facts, or as 
the embodiments of long series of causes, is an open 
question, which can be determined only by the ability 
of the pupils to master the higher conception. 

3. Ethical. 

History is replete with ethical content which the 
teacher should avail himself of as means for the moral 
uplifting of the child. This is the chief aim of history 
teaching — not to satisfy a craving for gossip ; not to 



HISTORY. 107 

exercise the intellectual faculties ; but to instill a habit 
of right conduct through emulation, and a love of coun- 
try through appreciation of others' sacrifices and one's 
own duties and responsibilities. 

II. The idea of concentric circles aptly characterizes 
the nature of the relation which exists among the mate- 
rials for historical study outlined above. They are 
not mutually exclusive ; on the contrary, they either 
partly or wholly include one another. Biography deals 
with events, with cause and effect, refers to institutions, 
and insinuates ethical concepts, even while it lays stress 
upon the individual ; events are not isolated phenom- 
ena; institutions are not purposeless; nor are ethical 
principles self -creative. 

Method. 

I. Biography and Events. 

We become interested in historical characters or 
events, either when they coincide with our sentiments, 
or when they are viewed in their original environments. 
We become still more interested in them when both 
these conditions are realized. 

The teacher of biography and of narrative, then, 
must search out all the causes which have been instru- 
mental in giving the subject of the child's study a place 
in history, and then arrange the facts in proper per- 
spective. With the plan of the lesson clear in his 
mind, he makes his presentation, guided by the thought 
that the nearer biography and narrative approximate 



108 HISTORY. 

actual occurrences, the more lifelike the characters and 
the events are, the greater is the appeal to the intellect 
and to the feelings of the listener. 

But knowledge alone will not suffice the successful 
teacher; but back of it, interwoven with it, impelling 
it, must be enthusiasm — interest which has gathered a 
large amount of momentum. For it is the enthusiast 
only who can awaken in others feelings akin to his own. 
Such interest on the part of the teacher comes only 
with perfect mastery of the subject. Deep, earnest, 
and frequent study is essential to success in teaching 
history. 

Minor points : 

1. Mechanical aids, such as pictures, portraits, maps 
— everything that conduces to a reconstruction of the 
environment — should be employed. 

2. Each biography should represent an epoch. 

3. Before assigning a lesson for home study, it 
should be carefully explained by the teacher. 

4. Before presenting a new lesson, the preceding 
lesson should be reviewed in order to connect the 
events in the mind of the child. 

5. Events should be mastered by interesting repeti- 
tion. 

6. Important dates should be committed to memory 
to serve as bases for comparison. 

7- The child should be trained to recite the lesson 
without prompting or questioning. 



HISTORY. HI 

II. Cause and Effect. 

Biography and events, which were originally taught 
as facts, are now made to yield to a higher process — 
the recognition of cause and effect. But in order 
that the process may be of value as an intellectual 
discipline, the child himself should be led to draw the 
inferences. 

III. Ethics. 

The good, the true, the beautiful, cannot be awakened 
into motive forces by direct teaching. Child nature 
resents such an attempt ; while the lessons taught in- 
directly by biography, by events, and by institutions 
insinuate themselves into the mind and heart of the 
child and become part of his inner life — his character. 



IX. — CIVICS. 

Aim: Training in and for citizenship. 

The teaching process, in order to be successful, must 
take into consideration the child to be educated, the 
material for his education, and the teacher, who is the 
medium between the two, their relative importance 
depending upon the aim of the lesson. Civics, which 
seeks primarily to impress its principles upon the moral 
sense of the child, must necessarily give prominence to 
the character of the teacher. 

Fundamentals. 
1. The child. 

(a) The underlying civic virtue is obedience to 
constituted authority ; hence, train the 
child first to a habit of implicit obedi- 
ence. 

(5) Obedience should aim to a rational and vol- 
untary compliance to an inner authority ; 
hence, rationalize the child's knowledge of 
law and order, broaden and deepen his 
sense of duty, and give impulse to his 



conduct. 



112 



civics. 115 

O) In intellectual education, interest is a con- 
dition to knowledge ; in moral education, 
knowledge is a condition to interest. 

(d) The child is in possession of all the elemen- 
tary concepts which underlie this study 
through intercourse with those with whom 
he comes in contact long before he begins 
the study of civics. 

0) The principles underlying the teaching of 
other studies obtain here also: Faith in 
the concrete, reasoning, conviction, action, 
are the steps in the development of a civic 
character. 

2. Material, 

(a) The purposes of government. 

The material must be concrete and familiar — the 
home, as represented by the head of the family; the 
school, by the teacher ; the city, by the policeman and 
the fireman. 

(6) A knowledge of our form of government. 

The material should include the leading facts and 
the underlying principles of municipal, state, and fed- 
eral governments. 

(<?) The duties of citizens. 
A comparison with other forms of government in 



116 CIVICS. 

order to emphasize such peculiar institutions as suf- 
frage, primaries, and conventions. 

(d) Historical personages and events. 
3. The teacher. 

Only the teacher who is zealous in the cause of good 
citizenship can arouse and maintain the child's interest 
in this study, can make him appreciate the blessings of 
our free government, can induce him to feel that he 
owes certain duties to his citizenship which call for 
cheerful responses on his part. 

Method. 
Model Lesson on Primaries (Rights and Duties). 1 

1. Facts: 

(a) Definition. 

(5) Date. 
(Y) Location, 
(df) Participants. 

(e) Purpose. 

2. Relations : 

(a) To the character of the candidates. 

(6) To the character of the government. 
(V) To the moral tone of the community. 

(d) To the American idea of government. 

(e) To self-respect, and to respect of others. 

1 Morally, attendance upon primaries and voting are duties 
legally, they are not — the more's the pity! 



civics. 119 



3. Feelings : 

O) Free government is a heritage handed down 
to us by centuries of oppression, suffering, 
and bloodshed; hence this right involves 
a duty, for "Eternal vigilance is the price 
of liberty." 

(b) Neglect to perforin the duties of citizenship 
results in political rings formed for selfish 
purposes and dominated by the one-man 
power — a travesty on our boasted self- 
government ! 

(<?) Such conditions ought not to be tolerated by 
enlightened men, both as individuals, as 
members of society, and as Americans. 

4. Action : 

The teacher can only sow the seed ; he must leave 
the rest to the future. 



X. — ARITHMETIC. 

Aims : 

1. Utilitarian. 

2. Intellectual. 

G-eneral considerations : 

I. As a school study, arithmetic includes : 

(a) Examples which deal with the fundamental 

processes of numbers, as 2| -7- 1^. 

(b) Problems, or questions involving numbers, in 

which the nature of the processes must first 
be determined by reasoning, as, If 1J yards 
of cloth cost $2|, what will 1 yard cost ? 

IL The processes involved in examples may be 
learned in two ways : 

(a) Through authority, as when a child is told 
that to divide one fraction by another, he 
must invert the divisor and proceed as in 
multiplication. 
(7>) Through induction, or the process whereby 
general laws are obtained through the in- 
vestigation of particulars, as, for instance, 
when the child, in comparing the results 
of several divisions of fractions, formulates 
for himself the law of the division of 
fractions. 

120 



ARITHMETIC. 123 

III. Arithmetical induction may be obtained in two 
ways : 

(a) By investigating the results of several con- 
crete presentations, as, for example : In 
dividing f by f, the child chooses a piece 
of paper as a unit, measures off § of it, 
then f of it ; then shows by actual meas- 
urement that the result of dividing- ^ bv ^ 

is *£. 6 3 J 5 

(5) By abstract reasoning. Thus, the answer to 
the above might have been obtained by 
reasoning in this wise : Dividing | by I 
will give -L - as a quotient, and dividing f 
by | will give £ of ^ or -^ as a quotient. 

Fundamentals. 

1. The child's knowledge of the laws of arithmetic 
should be the result of his own inductions. 

2. Induction through the concrete is the proper 
method for fundamental processes, and it must precede 
the abstract method. 

3. Concrete presentation for the purpose of induc- 
tion is demanded only by the processes involving whole 
numbers and common fractions. 

4. Processes should be mastered before an attempt 
is made to apply them to the solution of problems. 

5. Oral arithmetic is for instruction and drill ; writ- 
ten arithmetic, for exercises that deal with large numbers. 

6. Induction should be preceded by preparation and 



124 ARITHMETIC. 

followed by application, making the whole a process of 
apperception. 

7. The type is of great value in arithmetic. 

Method. 

Fundamental operations — integers* 
Guide: Concrete presentation. 

Counting. 

1. Counting is regarded as the initial step in arith- 
metic teaching, because : 

(a) It is a process with which the child is already 
acquainted when he enters school. 

(5) It is the natural basis of the fundamental 
operations. 

2. The teaching progression : idea, word, symbol. 
Symbols are introduced thus early because they embody 
in concrete form the child's abstractions, and thereby 
assist in his further progress, and because of their 
practical utility. 

Addition. 
Counting (synthesis). 

1. To teach the combination, three and two are five. 
(a) Children are given, or are told to take, first, 
3 sticks, and then 2 sticks. Questions fol- 
low with a view to directing the pupils' 
attention to the two groups and their com- 
positions ; after which they are told to 



ARITHMETIC. 127 

count the number of sticks in both groups 
(all together) ; and to announce the result, 
first as 5 sticks ; later, as 3 sticks and 2 
sticks are 5 sticks ; and lastly, as 3 and 2 
are 5. 

(5) Represent the process and the result on the 
blackboard (symbol of formula). 

(c) Drill. 
2. 15 

23 
The child has on his desk a bundle of 10 splints, and 
to the right of it 5 splints, and below them a piece of 
paper on which he has written the number 15. After 
recognizing the figure which represents the bundle and 
the figure which represents the loose sticks, he adds 8 
splints to the 5 splints, and writes the 8 below the 5. 
Upon counting his loose splints, he finds that he has 
13 of them, 10 of which he binds into a bundle, leaving 
3 loose splints. The latter he represents by a 3 placed 
in the units column; the bundle he transfers to the left 
(the transfer being marked on the paper), and the two 
bundles he represents by the figure 2 placed in the 
tens column. 

Subtraction. 
Counting (analysis). 

oo 

-18 

17 



128 ARITHMETIC. 

The child arranges the splints as in addition; also 
the paper on which the bundles are represented by 3, 
and the loose splints by 5. He discovers that he can- 
not take away 8 splints when he has only 5 splints, but 
that he can take a bundle and transfer it as 10 single 
splints to the 5 splints. He has now 2 bundles and 15 
splints (a result which he marks on his paper). From 
the 15 splints he takes away 8 splints, leaving 7 splints 
(represented on the paper as 7 units) ; and from the 2 
bundles he takes 1 bundle, leaving as a remainder 1 
bundle (represented on the paper as 1 ten). 

Multiplication. 

Counting of equal numbers with reference to 
"times" (synthesis). 

To teach 2 

x 3 
6 
Each pupil arranges 3 groups, each containing 2 
objects, then counts the number of objects in the 3 
groups, and announces that the 3 groups, each contain- 
ing 2 objects, have in all 6 objects. Next, he writes 
three 2's in a column, adds them, and tells how many 
2's make 6. This knowledge he now represents by the 
formula (briefer by comparison than that of addition) 

2 
x 3 
6 
and commits to memory. 



ARITHMETIC. 131 

Division. 
Counting of equal numbers with reference to "times" 
(analysis). g | 15 

5 

Each pupil arranges his 15 objects into groups, each 
group containing 3 objects. He then counts the num- 
ber of the groups and announces that he separated or 
divided the 15 objects into 5 groups, each containing 
3 objects. He next writes a sufficient number of 3's in 
a column to add up to 15, or he subtracts successively 
a sufficient number of 3's from 15 to have no re- 
mainder. In either case he finds that the 3 has been 
involved 5 times. He now learns the formula 

3 [15 
5 
and commits its substance to memory. 

Common Fractions. 
Guides : 

1. Concrete Presentation. 

2. The Formal Steps. 

Method I. 
Addition. 

1. Preparation: Find fractional parts of whole 
numbers. 

2. Presentation: 1. Add \ and \. 

The pupil is given 6 objects, such as buttons, marbles, 
splints, etc., and required to find \ of them and \ of 



132 ARITHMETIC. 

them ; then, having found that J of the marbles = 3 
marbles, and ^ of the marbles = 2 marbles, with proper 
guidance he concludes that ^ of the marbles 4- J of the 
marbles = f- of the marbles — a conclusion which he 
later transforms through abstraction into J + J = §. 

If he proceeds likewise with the following, and comes 
to the conclusion that 

9 i _i_ i Jr 

A ' 2 + 5 — 10* 

°' 3 ^ 4 — 12' 

4. l_i_l_il 3 8 

<*. 2 + ¥ + 5 — 40* 

he has all the material necessary for the third step. 
3. Comparison. 

The Common Denominator. 





Analysis. 




:ample. 


Original denominators. 


Common denominators. 


1. 


2, 3 




6 


2. 


2, 5 




10 


3. 


3, 4 




12 


4. 


2, 4, 5 




40 1 



1 There is no valid reason for the custom of discontinuing thus early 
in favor of the least common denominator the idea that the common 
denominator is the product of the various denominators. The argu- 
ments in favor of the latter practice may be briefly stated as follows : 

1. It does not burden the pupil with too many processes at the very 
outset. 

2. It leaves the fractions intact. 

3. The attention at the very outset is directed to the aim rather 
than to the preparatory matter. 

4. There is an economy of energy and time — reduction of the 
answer is all that is needed. 



ARITHMETIC. 135 

4. Generalization : 

The Common Denominator is the product of the 
several denominators. 

The Numerators. 

If the pupil's attention is now directed to the equa- 
tions 1 = |, l = ^, I = ^, etc., a vague intimation of 
which he already has, he will eventually discover that 
the ratio existing between the denominators of each 
equation is the same as that existing between their 
numerators; nor will he find great difficulty in 
discovering and in formulating a rule for finding 
the new numerators — in other words, to gener- 
alize. 

5. Application (see Fundamental 6) : 

Processes and concepts treated incidentally in the fore- 
going discussion, or readily deducible from it. 

1. Reduction to higher or lower terms. 
Corollary (a). 

The value of a fraction is not altered when both 
of its terms are multiplied or divided by the 
same number. 

Corollary (6). 
Factoring. 

2. The need for a common denominator. 



136 



ARITHMETIC. 



Subtraction of Fractions. 

With the single exception of the actual process of 
subtraction, the mode of presenting this topic is the 
same as in addition. 

Multiplication of Fractions. 
Multiply I by 1 
Take 6 splints ; then 

-| of 6 splints = 3 splints. 

^ of 3 splints = 1 splint. 

Proceeding as in addition, the conclusion is reached 



that 




i 


X 


1 _ 

3 — 


1 
6* 




In the same manner, 














2 


X 


2 _ 
5 — 


A- 




Comparison : 




2 
5 


X 


4 _ 
6 — 


8 
30* 




Original numerators. 












Resulting numerators. 


1,1 












1 


2,2 












4 


2,4 












8 


Conclusion : 














Original denominators. 










Resulting denominators. 


2,3 












6 


3,5 












15 


5, 6 












30 


Generalization : 


To 


multiply 


one 


fraction by another 


fraction, 















ARITHMETIC. 



139 



Division of Fractions. 



Divide J by \. 
Take 6 marbles ; 



then 
J of 6 = 2. 

i of 6 = 3 i and 



Other examples: 



Comparison: 



1 _:_ 

3 • 



1 _ 2 

2 — 3 



1 _ 



»• 

8 

15* 
9 
8' 



1. 

2. 
3. 
4. 



-L. 1 = 2 

•2 3* 



t+* = f 



JL 

15* 



4-3—8' 

A few well-directed questions will reveal to the child 
that 8, the numerator of the answer to example 3, for 
instance, is the product of 4 and 2, and that 15, the 
denominator of the same answer, is the product of 3 
and 5. Likewise with the other examples. 

Generalization: The law governing the division of 
fractions. 

Application : 

1 Attention must be called at this point to the change of base which 
takes place in multiplication, but not in division of fractions. A com- 
parison of the two processes will make this point clear. 

Multiply |x |. I Divide \ by |. 

i of 6 [base] =2. \ of 6 [base] = 2. 

\ of 2 [new base] = 1. | of 6 [same base] = 3. 

.-. * x i = *. .'.*•*- 1 = i 



140 



ARITHMETIC. 



Method II. 

(For more advanced pupils.) 

The presentation step only will be discussed here. 
The rest have already been discussed in Method I. 



1 + 1 



Addition. 




AD is the unit, AF is \ of it, and EH is J of it, 
and the question becomes: What part of AD are 
AF+EHl 

Assuming GrD 1 as the standard of comparison, or 
the unit of measure, it will be found by actual meas- 
urement that AF=S x GD, and EH =2 x GD; or 
that GrD is contained 2 times in EH, and 3 times in 
AF. AD is thus seen to be divided into 6 equal parts, 
of which AF contains 3 and EH, 2. 

Sothatl + i = f + f = f. 

1 In case the figure contains no exact unit of measure, one can 
be obtained by dividing its smallest division into 2 parts, or 3 parts, 
etc. 



1 — 1 



AKITHMETIC. 
Subtraction. 



143 



A ( 


3 E 






c 




mi 

iiiiiiii 









B H F D 

AD is the unit, AF is -| of it, tUT is J of it. 
The difference between AF and ^.5" is evidently GF, 
and the question becomes: What part of AD is GF? 

Assuming GF as the unit of measure, and proceed- 
ing as in addition, we have \ — i = # — f = J. 



xi 



Multiplication. 



A < 


gHl 








c 



















B H F D 

AD is the unit, AF is \ of it, and J.iZ is \ of ^l^, 1 
and the question becomes: What part of AD is AH? 
Evidently J. 



So that -i- x ^ = l 



1 See note to Method I. 




B H F D 

AD is the unit, AF is \ of it, AH is \ of it, and the 
question becomes : How many times is AH contained 
in AF, or how does AF compare in size to AH, or what 
part of AH is AF? 

Assuming GF as the unit of measure, and proceed- 
ing as previously, we find that AF is composed of 3 
units, and AH of 2 units ; so that AF is f of AH, or 
AH is contained 1J times in AF. 
Hence ^ -s- ^ = §. 



Numeration and Notation of Decimals. 

Guide : Common fractions. 

Preparation : The number of places an integer of 
any given value requires for its notation. 

Presentation: Notation of decimals, although it de- 
pends upon the device of place, can still be made the 
subject of inference, after the first few initial steps have 
been mastered by the child. 

For instance, if a child has learned to recognize and 
to write such fractions as .3, .04, .45, he can be led to 



ARITHMETIC. 147 

recognize the generalizations of both processes and to 
apply them. Thus he has already learned : 

1. *The denominator of a decimal fraction is indi- 
cated by a point before the numerator. 

2. That the device of place indicating the cipher is 
the same as in whole numbers. 

3. That numbers are regarded as increasing toward 
the left and vice versa in a tenfold ratio. 

4. That, counting the decimal point as a "place," 
there are just as many places needed to write a decimal 
fraction as a whole number of like character. 

Application : 

Express decimally four thousand ninety-eight hun- 
dred thousandths. 
Explanation : 

Since hundred-thousandths requires 6 places, and four 
thousand ninety-eight takes up 4 of the places, there 
still remain 2 places which should be filled by the deci- 
mal point and a cipher. Hence, beginning with the 
decimal point, the fraction is expressed as 

.04098. 
The method proposed here is recommended because 
(a) It makes no distinction between whole numbers 
and decimal fractions in the matter of place 
value. 
(5) It starts with the decimal point in notation — of 
great value when fractions are dictated for the 
purposes of addition or subtraction. 



148 ARITHMETIC. 



Reduction of Decimals. 



Reduce f to thousandths. 



| is the same as 3 -s- 8. • 

o . o— 10 . o— 10Q . o— 1000 . o— 1000 , 

.375 



or, decimally, 



8|3.000* 



Multiplication of Decimals. 

Decimals being merely a device, the laws governing 
its fundamental operations should be taught by refer- 
ence to common fractions, thus : 

- 1 - * J- — -1- • 1 y 1 - 01 
10 A 10 ~~ 100' * • * x A #i *~ ■ J± ' 

24 y 9 _ 216 . 91y Q_ 91 fi P f P 

TOO A 10 — 1000* ' • •*'**•*' — .^J-U, etu. 

Hence the rule : Point off in the product as many 
figures from the right as there are decimal figures in 
the multiplicand and the multiplier. 

Division of Decimals. 
(When the divisor is an integer.) 
.3 

X ' 10 • °— 10 X 3 _ 10* ■ • °|.» Qg 

9 9 _. q __ _a_ v l — .3 . arnci* 

a. 100 . o — 100 a 3 — 100 . . . oj.uy .0003 

3' 10000 "*" ** = to~tIto x i = TFooo- •"• 3|.0009 

Conclusion : The number of decimal places in the 
quotient is the same as the number of decimal places in 
the dividend. 









ARITHMETIC. 151 

(When the divisor is a fraction.) 

1. 9-s-^ = 9 x-V - = - 9 /=90-3. 

... .3[9= 3 [90. 

2. 9 -s- T f o = 9 x !$*■ = ■*£* = 900 -*- 3. 

... .03[9 = 3|900. 

3. 9 -s- T -3_ = 9 x Ujp = -^ = 9000 - 3. 

.-. .003[9 = 3|9000. 

Conclusion : We can get rid of a fractional divisor 
by multiplying both the divisor and the dividend by 
the denominator of the divisor. 

Deduction: Divide .009 by .03. g 

.03|.009 = 3[79 ; or, briefly, x 03. |x 00.9* 

Proof • - 9 - - 3_ — 9 v 10.0 _ _3_ 

x iuui . 1000 • ioo _ 1000 A 3 — 10* 

Percentage. 

Some of the difficulties which children encounter here 
will be obviated by defining " rate per cent " as so many 
out of each hundred. Thus 6 fo means 6 out of a hun- 
dred ; Qfo of 1500 means 1 6 out of each 1100 ; If of 
800 sheep means 7 sheep out of each group of 100 sheep. 

Analysis is the basic method in percentage. 

Examples : 

1. Find 5fo of 1800. 

1J6 of $800 = $8. 

5% of 1800 = 140, 



152 ARITHMETIC. 

or $8.00 

5 



$40.00 
2. What per cent of 200 is 8 ? 
\f of 200 = 2. 

Interest. 

With the 360 days method, the new factor presents 
but little difficulty. 

Example : 

Interest on 1500 for 1 yr. 2 mos. 12 days at 4±fo. 
$500 x432 x9 = 
360 x 200 

Proportion. 

Problems included under this topic should be solved 
by analysis, as follows : 

If 4 men, working 9 hrs. a day, require 2 days to 
build a wall 18 ft. high, how many days will 6 men, work- 
ing 8 hrs. a day, require to build a wall 12 ft. high ? 

Analysis : 

(a) If 4 men require 2 days, 

1 man requires 4x2 days, 

6 men require - of 4 x 2 days, 

or 4x2, 

r - days. 



ARITHMETIC. 155 

4 x 9 

(b) If 6 men require — — days to build 18 ft., 

1 4x2 

they will require — of — — days to build 
18 6 

1 ft. and 12 x-^- of i£? days to build 

18 6 J 

12 ft. 

12 x 4 x 2 , 

or ~— — days. 

18 x 6 J 

(c) If 6 men, building a wall 12 ft. and working 

12x4x2 
9 hrs. a day, require — - - — — - days, 

if they work 1 hr. a day, they will require 

Q 12x4x2 , 

9 X "18^6- dayS ' 

and working 8 hrs. a day, they will require 

1 , 12x4x2 , 

8 0f9x -18^T dayS ' 

or ^12x4_x_2 

8 x 18 x 6 J 

Square Root. 
First Step : Preparation. 

(a) The notion of square root is obtained by such 
exercises as 3x3=9, 9x9 = 81, 12x12 = 144, in 
which the product of the two equal factors is the 
square, and one of the two equal factors is the square 
root. Square and square root are thus seen to be rela- 



156 



ARITHMETIC. 



tive terms. Square root cannot be taught without 
reference to its square, nor can the latter be conceived 
apart from the former. 

(5) The symbols, 5 2 , V25. 

(c) The terms " power," " involution," " evolution," 
" index," "radix." 

(d) The analytic method of squaring numbers. As 
this topic does not engage the pupil's attention before 
he has reached a point in his mental development when 
he is already in possession of the elementary knowledge 
that underlies its elaboration, or before he can interpret 
the words of his teacher in terms of individualized or 
experienced notions, this method is preferable to the 
synthetic. 

The process of squaring any number, as, for ex- 
ample, 25, may be presented under three different 
operations. 

oo 

Which is the ordinary method 
of multiplication. 



(*) 



25 




25 




125 
50 

625 


of 


25 


25 

25 = 5 x 5, or 5 2 


100 = 20 x 5 


100 = 20 x 5 


400 = 20 x 20 or 20 2 



625 



Which shows the opera- 
tion in full. 



ARITHMETIC. 



159 



00 



20 + 5 
20 4-5 



20 x 5 + 5 2 

20 2 + 20 x 5 

20 2 + 2(20 x5) + 5 2 =625 



Which differs from 
b only in expressing 
the -f- symbol. 



(d) Again, 



(0 Or 



25 2 = (23 + 2) 2 = 625 

23 + 2 

23 + 2 

23 x 2 + 2 2 

23 2 + 23 x2 

23 2 + 2(23x2) + 2 2 = 625 

25 2 =(7 + 18) 2 =625 
7 + 18 
7 + 18 



7 xl8 + 18 2 
7 2 + 7xl8 



7 2 + 2(7 xl8) + 18 2 -625 



That is, the square of any number = the square of 
any two numbers whose sum equals that number + 
twice the product of those numbers. 

(/) Lastly, (20 + 5) 2 =20 + 5 

20+5 
20 2 +20x5 

+ 20 x 5 + 5 2 

20 2 + 2(20x5) + 5 2 = 625 



160 ARITHMETIC. 

In other words, the validity of the process and of the 
answer is in nowise affected when the process of multi- 
plication begins with the numbers to the left of the 
plus sign. 

If, for the sake of brevity, we represent the number 
to the left of the plus sign by x, and the number to the 
right of it by y, the process of involution (when the 
number is composed of two component parts) may be 
represented as, 

O + y) 2 = x 2 + 2xy + y 1 . 

A good rule to follow is not to consider the prepara- 
tion completed unless the child recognizes, as the result 
of his own efforts : 

1. That evolution is primarily a process of division ; 

2. That the process of extracting the square root 
may begin either at the right or at the left ; 

3. That the algebraic expression is a symbolic repre- 
sentation of a general truth which he can demonstrate 
satisfactorily ; and 

4. That the number may be divided into periods 
containing any number of figures. 



ARITHMETIC. 163 

Example : 

Find the square root of 54756. 

Beginning at the right, and pointing off periods of 
one figure, we have, 

1. 2 + 5 + 20 + 41 + 166=234 

5,4,7,5,6 = ^ + 2^ + ^ 2 

4 = ^/2 

52 = x 2 + 2 xy 
2x2x5) + 5x5) 45 = y 2 

707 = x 2 + 2 xy 
2 x 7 x 20) + 20 x 20) 680 = y 2 

4027 = z 2 + 2zy 
2 x 27 x 41) + 41 x 41) 3895 = y 2 

50132 = z 2 + 2^ 
2 x 68 x 166) + 166 x 166) 50132 = x 2 + 2 xy 

2. Beginning at the right and pointing off periods 
of two figures: 

7 + 61 + 166 = 234 

5, 47, 56 = x 2 + 2 xy + if ~ 

49 = y 2 

±701 = x 2 + 2xy 
2x7x61) + 61 x61) 4575 = / 

50132 = x 2 + 2 xy 
2 x 68 x 166) + 166 x 166) 50132 = x 2 + 2 ay 



164 ARITHMETIC. 

Second Step : Presentation. 

The following method is then shown to be the 
simplest. 

Find the square root of 978121. 
+ 9 : 
+ 80 : 989 
900 : 
978121 
900x900 = 810000 



= 168121 
2 x 900 x 80) + 80 x 80) = 15040 

= 17721 
2x980x9) + 9x9)= 17721 

Third Step : Comparison. 

Comparison of several examples of like character to 
above will make evident the following facts : 

1. That, when a quantity is expressed by an even 
number of figures, its square root contains one half its 
number of figures ; but when a quantity is expressed 
by an odd number of figures, its square root con- 
sists of one more than one half its number of figures. 
This principle may also be demonstrated by involution. 

2. That the subtrahend (with the possible exception 
of the initial and the final) contains an even number 
of zeros. 

3. That (with the single exception of the initial sub- 
trahend, which involves either one or two figures, de- 






ARITHMETIC. 167 

pending upon whether the square is made up of an 
even or of an odd number of figures) two figures of the 
minuend are successively involved in finding the square 
root corresponding to the given period. 

4. That the sum of the products of two multiplica- 
tions, both having the same number as a multiplier, 
equals the product of the sum of the multiplicands by 
the multiplier ; as, for example, 

9x5 + 5x5 = 14 x 5. 

Fourth Step : Generalization. 

The aim of the lesson is best served by leading the 
child to frame rules, rather than to allow him to memo- 
rize the rules found in the text-book. 

Fifth Step : Application. 

(V) Find the square root of 54756 and of numerous 
other examples as follows : 

2 3 4 
5, 47, 56 
4_ 
43) 147 
129 
464) 1856 
1856 

(5) Solve the following and similar problems : 

A man owns a farm in the form of a square which 

contains 45 A. 25 sq. rd. How many rods in length or 

breadth is it ? 



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